Atlas › Modular forms › Weight 2 › Level 771 › 771.2.a › 771.2.a.e

Newform orbit 771.2.a.e

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$14q - 2q^{2} - 14q^{3} + 20q^{4} + 9q^{5} + 2q^{6} + 4q^{7} - 3q^{8} + 14q^{9} + 10q^{10} - 2q^{11} - 20q^{12} + 17q^{13} + 6q^{14} - 9q^{15} + 28q^{16} + 2q^{17} - 2q^{18} + 16q^{19} + 12q^{20} - 4q^{21} + 8q^{22} - 23q^{23} + 3q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{14} - 7x^{13} - 60x^{12} + 455x^{11} + 1152x^{10} - 10655x^{9} - 6824x^{8} + 112886x^{7} - 22144x^{6} - 556529x^{5} + 314495x^{4} + 1258461x^{3} - 940893x^{2} - 1058036x + 901802$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
257	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	−2	−14	9	4	−2	17	2	16	−23	29	17	24	21	6	−5	−2	4	82	13	8	30	35	−21	28	15

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.243 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound. The label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 3, 257, 771]: [0, 1, 22, 88] (certified at ell = 2147483137)
T = [(2, 1), (5, 1)]; h of degree 43 = 1 x 43 (phi(ord eps) x dim S_k^new); 5 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(771, 2)[4]   # 771.2.a.e

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 771:2 1000